Bessel functions

Explicit plots of the $J_{0}$ and $J_{1}$ from

Complex map of $J_{1}$ by ; $u+\mathrm {i} v=J_{1}(x+\mathrm {i} y)$

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Bessel functions are solutions of the Bessel differential equation:

$z^{2}{\frac {d^{2}w}{dz^{2}}}+z{\frac {dw}{dz}}+(z^{2}-\alpha ^{2})w=0$

where α is a constant.

Because this is a second-order differential equation, it should have two linearly-independent solutions:

(i) J_α(x) and
(ii) Y_α(x).

In addition, a linear combination of these solutions is also a solution:

(iii) H_α(x) = C₁ J_α(x) + C₂ Y_α(x)

where C₁ and C₂ are constants.

These three kinds of solutions are called Bessel functions of the first kind, second kind, and third kind.

Properties

Many properties of functions $J$, $Y$ and $H$ are collected at the handbook by Abramowitz, Stegun .

Integral representations

$\!\!\!\!\!\!\!\!\!\!(9.1.20)~~~\displaystyle J_{\nu }(z)={\frac {(z/2)^{\nu }}{\pi ^{1/2}~(\nu -1/2)!}}~\int _{0}^{\pi }~\cos(z\cos(t))\sin(t)^{2\nu }~t~\mathrm {d} t$

Expansions at small argument

$\displaystyle J_{\alpha }(z)=\left({\frac {z}{2}}\right)^{\!\alpha }~\sum _{k=0}^{\infty }~{\frac {(-z^{2}/4)^{k}}{k!~(\alpha \!+\!k)!}}$

The series converges in the whole complex $z$ plane, but fails at negative integer values of $\alpha$ . The postfix form of factorial is used above; $k!=\mathrm {Factorial} (k)$.

Applications

Bessel functions arise in many applications. For example, Kepler’s Equation of Elliptical Motion, the vibrations of a membrane, and heat conduction, to name a few. In paraxial optics the Bessel functions are used to describe solutions with circular symmetry.